Theorem diagcl

Description: The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor ( y e. D |-> X ) is a construction that is natural in X (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017) (Revised by Mario Carneiro, 15-Jan-2017)

	Ref	Expression
Hypotheses	diagval.l	⊢ 𝐿 = ( 𝐶 Δfunc 𝐷 )
	diagval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	diagval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
	diagcl.q	⊢ 𝑄 = ( 𝐷 FuncCat 𝐶 )
Assertion	diagcl	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐶 Func 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		diagval.l	⊢ 𝐿 = ( 𝐶 Δfunc 𝐷 )
2		diagval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
3		diagval.d	⊢ ( 𝜑 → 𝐷 ∈ Cat )
4		diagcl.q	⊢ 𝑄 = ( 𝐷 FuncCat 𝐶 )
5	1 2 3	diagval	⊢ ( 𝜑 → 𝐿 = ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐶 1stF 𝐷 ) ) )
6		eqid	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐶 1stF 𝐷 ) ) = ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐶 1stF 𝐷 ) )
7		eqid	⊢ ( 𝐶 ×c 𝐷 ) = ( 𝐶 ×c 𝐷 )
8		eqid	⊢ ( 𝐶 1stF 𝐷 ) = ( 𝐶 1stF 𝐷 )
9	7 2 3 8	1stfcl	⊢ ( 𝜑 → ( 𝐶 1stF 𝐷 ) ∈ ( ( 𝐶 ×c 𝐷 ) Func 𝐶 ) )
10	6 4 2 3 9	curfcl	⊢ ( 𝜑 → ( ⟨ 𝐶 , 𝐷 ⟩ curryF ( 𝐶 1stF 𝐷 ) ) ∈ ( 𝐶 Func 𝑄 ) )
11	5 10	eqeltrd	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐶 Func 𝑄 ) )